{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2018:CGYYCLTARQDUHFW7UYV23OP62F","short_pith_number":"pith:CGYYCLTA","schema_version":"1.0","canonical_sha256":"11b1812e608c074396dfa62badb9fed16481d14353b511b32939dd8afaebc55c","source":{"kind":"arxiv","id":"1810.09748","version":1},"attestation_state":"computed","paper":{"title":"A covariance equation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CV","authors_text":"El Hassan Youssfi","submitted_at":"2018-10-23T09:48:02Z","abstract_excerpt":"Let $\\S$ be a commutative semigroup with identity $e$ and let $\\Gamma$ be a compact subset in the pointwise convergence topology of the space $\\S'$ of all non-zero multiplicative functions on $\\S.$ Given a continuous function $F: \\Gamma \\to \\mathbb C$ and a complex regular Borel measure $\\mu$ on $\\Gamma$ such that $\\mu(\\Gamma) \\not = 0.$ It is shown that $$ \\mu(\\Gamma) \\int_{\\Gamma} \\varrho(s) \\overline{\\varrho(t)} |F|^2(\\varrho) d\\mu(\\varrho) = \\int_{\\Gamma} \\varrho(s) F(\\varrho) d\\mu(\\varrho) \\int_{\\Gamma} \\overline{\\varrho(t) F(\\varrho)} d\\mu(\\varrho) $$ for all $(s, t) \\in \\S\\times \\S$ if "},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1810.09748","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CV","submitted_at":"2018-10-23T09:48:02Z","cross_cats_sorted":[],"title_canon_sha256":"b0d713e6e7122244bd79ca2a43e7902d30cb9b1cc254a9674844d4183029cb3b","abstract_canon_sha256":"aaa9a830b55d77fe6912cb5c40350acc0155d95d810668896c7b291c69293827"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:02:29.525095Z","signature_b64":"1tSwnS9mgvoRgqFsx0c3M3AyqhLtRatFCaRbgWwAUePxGheBIfM8zeFo4tk9yLjDB+zSxcFthPxoAKt6YVhJCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"11b1812e608c074396dfa62badb9fed16481d14353b511b32939dd8afaebc55c","last_reissued_at":"2026-05-18T00:02:29.524384Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:02:29.524384Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A covariance equation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CV","authors_text":"El Hassan Youssfi","submitted_at":"2018-10-23T09:48:02Z","abstract_excerpt":"Let $\\S$ be a commutative semigroup with identity $e$ and let $\\Gamma$ be a compact subset in the pointwise convergence topology of the space $\\S'$ of all non-zero multiplicative functions on $\\S.$ Given a continuous function $F: \\Gamma \\to \\mathbb C$ and a complex regular Borel measure $\\mu$ on $\\Gamma$ such that $\\mu(\\Gamma) \\not = 0.$ It is shown that $$ \\mu(\\Gamma) \\int_{\\Gamma} \\varrho(s) \\overline{\\varrho(t)} |F|^2(\\varrho) d\\mu(\\varrho) = \\int_{\\Gamma} \\varrho(s) F(\\varrho) d\\mu(\\varrho) \\int_{\\Gamma} \\overline{\\varrho(t) F(\\varrho)} d\\mu(\\varrho) $$ for all $(s, t) \\in \\S\\times \\S$ if "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1810.09748","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"aliases":[{"alias_kind":"arxiv","alias_value":"1810.09748","created_at":"2026-05-18T00:02:29.524505+00:00"},{"alias_kind":"arxiv_version","alias_value":"1810.09748v1","created_at":"2026-05-18T00:02:29.524505+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1810.09748","created_at":"2026-05-18T00:02:29.524505+00:00"},{"alias_kind":"pith_short_12","alias_value":"CGYYCLTARQDU","created_at":"2026-05-18T12:32:16.446611+00:00"},{"alias_kind":"pith_short_16","alias_value":"CGYYCLTARQDUHFW7","created_at":"2026-05-18T12:32:16.446611+00:00"},{"alias_kind":"pith_short_8","alias_value":"CGYYCLTA","created_at":"2026-05-18T12:32:16.446611+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/CGYYCLTARQDUHFW7UYV23OP62F","json":"https://pith.science/pith/CGYYCLTARQDUHFW7UYV23OP62F.json","graph_json":"https://pith.science/api/pith-number/CGYYCLTARQDUHFW7UYV23OP62F/graph.json","events_json":"https://pith.science/api/pith-number/CGYYCLTARQDUHFW7UYV23OP62F/events.json","paper":"https://pith.science/paper/CGYYCLTA"},"agent_actions":{"view_html":"https://pith.science/pith/CGYYCLTARQDUHFW7UYV23OP62F","download_json":"https://pith.science/pith/CGYYCLTARQDUHFW7UYV23OP62F.json","view_paper":"https://pith.science/paper/CGYYCLTA","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1810.09748&json=true","fetch_graph":"https://pith.science/api/pith-number/CGYYCLTARQDUHFW7UYV23OP62F/graph.json","fetch_events":"https://pith.science/api/pith-number/CGYYCLTARQDUHFW7UYV23OP62F/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/CGYYCLTARQDUHFW7UYV23OP62F/action/timestamp_anchor","attest_storage":"https://pith.science/pith/CGYYCLTARQDUHFW7UYV23OP62F/action/storage_attestation","attest_author":"https://pith.science/pith/CGYYCLTARQDUHFW7UYV23OP62F/action/author_attestation","sign_citation":"https://pith.science/pith/CGYYCLTARQDUHFW7UYV23OP62F/action/citation_signature","submit_replication":"https://pith.science/pith/CGYYCLTARQDUHFW7UYV23OP62F/action/replication_record"}},"created_at":"2026-05-18T00:02:29.524505+00:00","updated_at":"2026-05-18T00:02:29.524505+00:00"}